Geometry — Triangles, Quadrilaterals, Congruence and Similarity
Overview
Geometry forms a substantial portion of the Mathematics section in Bihar TET Paper II. This topic tests both conceptual understanding and problem-solving ability — expect questions on properties of triangles and quadrilaterals, congruence criteria, similarity theorems, and area relationships. Mastery here directly helps in mensuration problems as well.
For Bihar TET, focus on the standard results: angle-sum properties, congruence rules (SSS, SAS, ASA, AAS, RHS), similarity criteria (AA, SSS, SAS), and the Basic Proportionality Theorem (Thales theorem). Questions typically involve finding unknown angles, proving triangles congruent or similar, and applying properties to calculate lengths or areas. A clear grasp of definitions and theorems, combined with practice on standard problem types, ensures scoring in this section.
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Key Concepts
- **Triangle angle-sum property**: The sum of interior angles of any triangle is 180°. The exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
- **Types of triangles**: Classified by sides (scalene, isosceles, equilateral) and by angles (acute, right, obtuse). An equilateral triangle has all angles equal to 60°.
- **Quadrilateral angle-sum property**: The sum of interior angles of any quadrilateral is 360°.
- **Special quadrilaterals hierarchy**: Square → Rectangle → Parallelogram → Quadrilateral; Square → Rhombus → Parallelogram. A square has all properties of both rectangle and rhombus.
- **Congruence of triangles**: Two triangles are congruent if they have exactly the same shape and size — all corresponding sides and angles are equal.
- **Similarity of triangles**: Two triangles are similar if they have the same shape but not necessarily the same size — corresponding angles are equal and corresponding sides are in proportion.
- **Basic Proportionality Theorem (BPT / Thales theorem)**: If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
- **Area relationship in similar triangles**: If two triangles are similar with sides in ratio k, then their areas are in ratio k².
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Formulas / Key Facts
**Triangle Properties**
- Sum of angles = 180°
- Exterior angle = Sum of two remote interior angles
- In a right triangle: hypotenuse² = base² + perpendicular² (Pythagoras theorem)
**Quadrilateral Properties**
- Sum of angles = 360°
- Parallelogram: opposite sides equal and parallel, opposite angles equal, diagonals bisect each other
- Rectangle: all angles 90°, diagonals equal
- Rhombus: all sides equal, diagonals bisect at 90°
- Square: all sides equal, all angles 90°, diagonals equal and bisect at 90°
- Trapezium: one pair of opposite sides parallel
**Congruence Criteria (5 rules)**
- SSS (Side-Side-Side): All three sides equal
- SAS (Side-Angle-Side): Two sides and included angle equal
- ASA (Angle-Side-Angle): Two angles and included side equal
- AAS (Angle-Angle-Side): Two angles and one non-included side equal
- RHS (Right angle-Hypotenuse-Side): For right triangles — hypotenuse and one side equal
**Similarity Criteria (3 rules)**
- AA (Angle-Angle): Two angles of one triangle equal to two angles of another
- SSS (Side-Side-Side ratio): All three pairs of sides in the same ratio
- SAS (Side-Angle-Side ratio): Two pairs of sides in proportion and included angles equal
**Thales Theorem (BPT)**
- If DE || BC in triangle ABC, then AD/DB = AE/EC
**Area Ratio in Similar Triangles**
- If triangle ABC ~ triangle PQR with ratio of sides = k, then Area(ABC)/Area(PQR) = k²
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Worked Examples
**Example 1: Finding an angle in a triangle**
In triangle PQR, angle P = 65° and angle Q = 45°. Find angle R and the exterior angle at R.
*Solution:*
- Sum of angles = 180°
- Angle R = 180° − 65° − 45° = 70°
- Exterior angle at R = angle P + angle Q = 65° + 45° = 110°
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**Example 2: Proving triangles congruent**
In triangles ABC and DEF, AB = DE = 5 cm, BC = EF = 7 cm, and angle B = angle E = 60°. Are they congruent?
*Solution:*
- We have two sides equal: AB = DE, BC = EF
- The included angle between these sides: angle B = angle E
- By SAS criterion, triangle ABC ≅ triangle DEF
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**Example 3: Applying Basic Proportionality Theorem**
In triangle ABC, DE is parallel to BC. If AD = 4 cm, DB = 6 cm, and AE = 5 cm, find EC.
*Solution:*
- By BPT: AD/DB = AE/EC
- 4/6 = 5/EC
- EC = (5 × 6)/4 = 30/4 = 7.5 cm
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**Example 4: Area ratio in similar triangles**
Triangle ABC is similar to triangle PQR. If AB = 6 cm and PQ = 9 cm, and area of triangle ABC = 48 cm², find area of triangle PQR.
*Solution:*
- Ratio of corresponding sides = 6/9 = 2/3
- Ratio of areas = (2/3)² = 4/9
- Area(ABC)/Area(PQR) = 4/9
- 48/Area(PQR) = 4/9
- Area(PQR) = (48 × 9)/4 = 108 cm²
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Common Mistakes
- **Confusing congruence with similarity** → Congruent triangles are identical in size and shape; similar triangles have equal angles but sides only in proportion, not necessarily equal.
- **Using wrong congruence criteria (AAA or SSA)** → AAA proves similarity, not congruence. SSA (or ASS) is not a valid congruence rule — it can produce two different triangles.
- **Forgetting to check the included angle in SAS** → The angle must be between the two given sides. If the angle is not included, SAS does not apply.
- **Applying Pythagoras theorem to non-right triangles** → The formula a² + b² = c² works only when c is the hypotenuse of a right triangle.
- **Incorrect area ratio calculation** → For similar triangles, area ratio equals the square of the side ratio, not the side ratio itself. If sides are in ratio 2:3, areas are in ratio 4:9, not 2:3.
- **Misapplying BPT when line is not parallel** → Thales theorem requires the line to be parallel to one side. Always verify parallelism before using the proportion.
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Quick Reference
- **Triangle angles sum to 180°; quadrilateral angles sum to 360°.**
- **Congruence rules: SSS, SAS, ASA, AAS, RHS — remember AAA and SSA do NOT prove congruence.**
- **Similarity rules: AA, SSS (ratio), SAS (ratio).**
- **BPT: Line parallel to one side divides other two sides proportionally.**
- **Similar triangles: sides in ratio k ⇒ areas in ratio k².**
- **Square inherits all properties of rectangle AND rhombus.**